Metrical almost periodicity and applications

Abstract

In this paper, we analyze various classes of multi-dimensional almost periodic type functions in general metric. The main classes of functions under our consideration are (R, B, P, L)-multi-almost periodic functions, (R X , B, P, L)-multi-almost periodic functions, Bohr (B, I ′ , ρ, P)-almost periodic functions and (B, I ′ , ρ, P)-uniformly recurrent functions. We clarify the main structural properties for the introduced classes of almost periodic type functions and provide some applications of our results to the abstract Volterra integro-differential equations.

Metrical approximations of functions

“…Concerning the convolution invariance of function spaces introduced in this paper, we will only present here a few comments and a concrete application in the study of the existence and uniqueness of metrically Besicovitch almost periodic solutions of the heat equation in R n ; the theoretical analysis is very similar to the analysis of the invariance under the actions of infinite convolution products carried out in the next subsection (see [21,22,23] for some results obtained recently in this direction).…”

“…The main aim of Proposition 2.8 is to indicate that the notion of (φ, R, B, φ, F, P)-normality is very general as well as that any function belonging to the introduced function space e − (B, φ, F) − B P• (Λ × X : Y ) [e − (B, φ, F) j∈Nn − B P• I (Λ × X : Y )] is (φ, R, B, φ, F, P)-normal under certain logical assumptions (cf. also Proposition 2.9, where we reconsider the statements of [21,Proposition 3.7,Corollary 3.8] in our new framework).…”

“…The usual notion of (R, B, P)-normality is obtained by plugging The uniformly convergent sequences of almost periodic type functions and the behaviour of its limit function have been examined in many structural results established so far (see, e.g., [21,Proposition 2.7] for such a result regarding metrically almost periodic type functions). In this paper, we will clarify only one result concerning this issue:…”

“…The first systematic study of metrical almost periodicity was conducted by the second named author in 2021 ( [21]). The Stepanov, Weyl and Besicovitch classes of metrical almost periodic functions were considered in [22], [23] and [24], respectively.…”

Multi-dimensional Besicovitch almost periodic type functions and applications

“…Besides many other topics, in this paper we have not reconsidered the notions of M p -almost periodicity and M p -regularity in the multi-dimensional setting as well (see [3] and [7] for more details about the subject). The class of metrical multi-dimensional Besicovitch almost periodic functions will be considered somewhere else following the general approach obeyed in [45].…”